We are tasked to solve the equation:

$\log_3 (2x + 1) = \log_7 49$

Step 1: Simplify $\log_7 49$

We know that $49 = 7^2$. Using the logarithmic property $\log_b b^k = k$, we get:

$\log_7 49 = \log_7 (7^2) = 2$

So the equation simplifies to:

$\log_3 (2x + 1) = 2$

Step 2: Rewrite the logarithmic equation in exponential form

The property of logarithms, $\log_b a = c \implies b^c = a$, allows us to rewrite the equation as:

$3^2 = 2x + 1$

This simplifies to:

$9 = 2x + 1$

Step 3: Solve for $x$

Subtract 1 from both sides:

$9 - 1 = 2x$

$8 = 2x$

Divide both sides by 2:

$x = 4$

Final Answer:

$x = 4$

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Would you like further explanation or details about the steps?

Related Questions:

1. How do logarithmic properties simplify equations?
2. What are the different forms of logarithmic equations?
3. How can exponential and logarithmic functions be graphed?
4. What are the rules of converting between logarithmic and exponential forms?
5. Can logarithmic equations have no solutions or multiple solutions?

Tip:

Always check if the solution satisfies the domain of the logarithmic function (e.g., the argument must be positive).